Finite Group Databasebeta

← C53×#12096 #27219608

Subgroups — C53×#12096 #27219608

Order 13568.

726 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2679C211yesmaximal, characteristic, complemented
2C53×#2006C211yesmaximal, characteristic, complemented
3C53×#1598C212yesmaximal, complemented
4C53×#1345C212yesmaximal, complemented
5C53×#1318C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C53—21no—
7C53×#537C2211yescharacteristic, complemented
8C2×#120×C53—22no—
9C2×#120×C53—21no—
10C53×#447—22no—
11C2×#104×C53—21no—
12C53×#394—22no—
13C53×#394C2212yes—
14C53×#390—21no—
15C53×#390C2211yescharacteristic
16C53×#343C2211yescharacteristic, complemented
17C53×#327—22no—
18C53×#324C2212yescomplemented
19C25×C53—21no—
20C2×#41×C53—21no—
21C22×D4×C53—21no—
22C22×D4×C53—22no—
23C22×D4×C53D411yescharacteristic
24C22×D4×C53D412yes—
25AGL(1,ℤ/8)×C53—42no—
26AGL(1,ℤ/8)×C53—42no—
27C2×SD16×C53—41no—
28C2×D8×C53—41no—
29C2×D8×C53—42no—
30C2×M16×C53—21no—
31C22⋊Q8×C53—41no—
32#121×C53—22no—
33#121×C53—41no—
34#120×C53—48no—
35#120×C53—42no—
36#120×C53—42no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C42×C53D411yescharacteristic
45#104×C53—42no—
46#102×C53—42no—
47C22⋊C8×C53—22no—
48#95×C53D412yes—
49C24×C53—22no—
50C24×C53—21no—
51C24×C53—44no—
52C24×C53—42no—
53C24×C53—82no—
54C24×C53—81no—
55C24×C53—81no—
56#41×C53—41no—
57#41×C53—41no—
58#41×C53—41no—
59#41×C53—41no—
60C2×Q8×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—24no—
63C2×D4×C53—22no—
64C2×D4×C53—21no—
65C2×D4×C53—42no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53—44no—
69C2×D4×C53—44no—
70C2×D4×C53—44no—
71C2×D4×C53—41no—
72C2×D4×C53—81no—
73C2×D4×C53—81no—
74C2×D4×C53—82no—
75C2×D4×C53D812yescomplemented
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D412yes—
81SD16×C53—44no—
82D8×C53—44no—
83D8×C53—84no—
84M16×C53—22no—
85M16×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—42no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—81no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—44no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—42no—
98C42×C53—41no—
99C23×C53—24no—
100C23×C53—22no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—41no—
108C23×C53—42no—
109C23×C53—41no—
110C23×C53—42no—
111C23×C53—41no—
112C23×C53—88no—
113C23×C53—84no—
114C23×C53—88no—
115C23×C53—84no—
116C23×C53—84no—
117C23×C53#12011yescharacteristic
118Q8×C53—41no—
119Q8×C53—41no—
120D4×C53—42no—
121D4×C53—41no—
122D4×C53—41no—
123D4×C53—44no—
124D4×C53—81no—
125D4×C53—84no—
126D4×C53—82no—
127D4×C53—82no—
128D4×C53—88no—
129D4×C53—81no—
130D4×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C8×C53—42no—
147C8×C53—82no—
148#12096C5311yesmaximal, characteristic, direct product factor, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—44no—
152C22×C53—44no—
153C22×C53—44no—
154C22×C53—42no—
155C22×C53—44no—
156C22×C53—82no—
157C22×C53—82no—
158C22×C53—82no—
159C22×C53—816no—
160C22×C53—82no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—82no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53#44711yescharacteristic
168C22×C53#45311yescharacteristic
169C22×C53B(4,𝔽2)11yescharacteristic
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—42no—
173C4×C53—42no—
174C4×C53—82no—
175C4×C53—81no—
176#2679C2×C5311yescharacteristic, complemented
177#2006C2×C5311yescharacteristic, complemented
178#1598C2×C5312yescomplemented
179#1345C2×C5312yescomplemented
180#1318C2×C5311yescharacteristic, complemented
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—82no—
184C2×C53—84no—
185C2×C53—161no—
186C2×C53#159511yescharacteristic
187C2×C53#219611yescharacteristic
188C2×C53#220211yescharacteristic
189C2×AGL(1,ℤ/8)—21no—
190#537C22×C5311yescharacteristic, complemented
191C2×#120—22no—
192C2×#120—21no—
193#447—22no—
194C2×#104—21no—
195#394—22no—
196#394C22×C5312yes—
197#390—21no—
198#390C22×C5311yescharacteristic
199#343C22×C5311yescharacteristic, complemented
200#327—22no—
201#324C22×C5312yescomplemented
202C53#1209611yescharacteristic, direct product factor, Sylow
203C25—21no—
204C2×#41—21no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4D4×C5311yescharacteristic
208C22×D4D4×C5312yes—
209AGL(1,ℤ/8)—42no—
210AGL(1,ℤ/8)—42no—
211C2×SD16—41no—
212C2×D8—41no—
213C2×D8—42no—
214C2×M16—21no—
215C22⋊Q8—41no—
216#121—22no—
217#121—41no—
218#120—48no—
219#120—42no—
220#120—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C23×C5311yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42D4×C5311yescharacteristic
229#104—42no—
230#102—42no—
231C22⋊C8—22no—
232#95D4×C5312yes—
233C24—22no—
234C24—21no—
235C24—44no—
236C24—42no—
237C24—82no—
238C24—81no—
239C24—81no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—81no—
257C2×D4—81no—
258C2×D4—82no—
259C2×D4D8×C5312yescomplemented
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4C2×D4×C5311yescharacteristic
264C22×C4C2×D4×C5312yes—
265SD16—44no—
266D8—44no—
267D8—84no—
268M16—22no—
269M16—41no—
270C2×C8—41no—
271C2×C8—42no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—41no—
283C23—24no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—88no—
297C23—84no—
298C23—88no—
299C23—84no—
300C23—84no—
301C23#120×C5311yescharacteristic
302Q8—41no—
303Q8—41no—
304D4—42no—
305D4—41no—
306D4—41no—
307D4—44no—
308D4—81no—
309D4—84no—
310D4—82no—
311D4—82no—
312D4—88no—
313D4—81no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C2×D8×C5311yescharacteristic
329C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
330C8—42no—
331C8—82no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22—816no—
343C22—81no—
344C22—82no—
345C22—81no—
346C22—82no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C53×#44711yescharacteristic
351C22C53×#45311yescharacteristic
352C22C53×B(4,𝔽2)11yescharacteristic
353C4—21no—
354C4—21no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C2—21no—
360C2—21no—
361C2—84no—
362C2—82no—
363C2—161no—
364C2C53×#159511yescharacteristic
365C2C53×#219611yescharacteristic
366C2C53×#220211yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C53×#2679C2semidirect
2. C53×#2006C2semidirect
3. C53×#1598C2semidirect
4. C53×#1345C2semidirect
5. C53×#1318C2semidirect
7. C53×#537C22semidirect
16. C53×#343C22semidirect
18. C53×#324C22semidirect
75. C2×D4×C53D8semidirect
148. #12096C53direct (N × H)
176. #2679C2×C53semidirect
177. #2006C2×C53semidirect
178. #1598C2×C53semidirect
179. #1345C2×C53semidirect
180. #1318C2×C53semidirect
190. #537C22×C53semidirect
199. #343C22×C53semidirect
201. #324C22×C53semidirect
202. C53#12096direct (N × H)
259. C2×D4D8×C53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C53×#26792
C53×#20062
C53×#15982
C53×#13452
C53×#13182
#1209653